Bounding Fitting Heights of Two Classes of Character Degree Graphs

2014 ◽  
Vol 21 (02) ◽  
pp. 355-360
Author(s):  
Xianxiu Zhang ◽  
Guangxiang Zhang
Keyword(s):  
Solvable Group ◽  
Disjoint Union ◽  
Character Degree ◽  
Finite Solvable Group ◽  
Total Degree ◽  
Character Degree Graph ◽  
Vertex Set ◽  
Fitting Height ◽  
Degree Graph

In this article, we prove that a finite solvable group with character degree graph containing at least four vertices has Fitting height at most 4 if each derived subgraph of four vertices has total degree not more than 8. We also prove that if the vertex set ρ(G) of the character degree graph Δ(G) of a solvable group G is a disjoint union ρ(G) = π1 ∪ π2, where |πi| ≥ 2 and pi, qi∈ πi for i = 1,2, and no vertex in π1 is adjacent in Δ(G) to any vertex in π2 except for p1p2 and q1q2, then the Fitting height of G is at most 4.

On the character degree graph of solvable groups. II. Discon- nected graphs

2001 ◽  
Vol 38 (1-4) ◽  
pp. 339-355 ◽  
Author(s):  
P. P. Pálfy
Keyword(s):  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Character Degree ◽  
Finite Solvable Group ◽  
Nilpotent Length ◽  
Character Degree Graph ◽  
Degree Graph

Let G be a finite solvable group and ¡ a set of primes such that the degree of each irreducible character of G is either a ¡-number or a ¡0-number. We show that such groups have a very restricted structure, for example, their nilpotent length is at most 4. We also prove that Huppert's ^{ÿ Conjecture is valid for these groups.

Character Degree Graphs of Solvable Groups of Fitting Height 2

2006 ◽  
Vol 49 (1) ◽  
pp. 127-133 ◽  
Author(s):  
Mark L. Lewis
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Character Degree ◽  
Character Degrees ◽  
Irreducible Characters ◽  
Vertex Set ◽  
Fitting Height ◽  
A Finite Group

AbstractGiven a finite group G, we attach to the character degrees of G a graph whose vertex set is the set of primes dividing the degrees of irreducible characters of G, and with an edge between p and q if pq divides the degree of some irreducible character of G. In this paper, we describe which graphs occur when G is a solvable group of Fitting height 2.

ON THE REGULARITY OF CHARACTER DEGREE GRAPHS

2019 ◽  
Vol 100 (3) ◽  
pp. 428-433 ◽  
Author(s):  
Z. SAYANJALI ◽  
Z. AKHLAGHI ◽  
B. KHOSRAVI
Keyword(s):  
Finite Group ◽  
Character Degree ◽  
Character Degrees ◽  
Prime Divisors ◽  
Character Degree Graph ◽  
Vertex Set ◽  
Degree Graph ◽  
A Finite Group ◽  
Regular Bipartite Graph ◽  
Complex Characters

Let $G$ be a finite group and let $\text{Irr}(G)$ be the set of all irreducible complex characters of $G$. Let $\unicode[STIX]{x1D70C}(G)$ be the set of all prime divisors of character degrees of $G$. The character degree graph $\unicode[STIX]{x1D6E5}(G)$ associated to $G$ is a graph whose vertex set is $\unicode[STIX]{x1D70C}(G)$, and there is an edge between two distinct primes $p$ and $q$ if and only if $pq$ divides $\unicode[STIX]{x1D712}(1)$ for some $\unicode[STIX]{x1D712}\in \text{Irr}(G)$. We prove that $\unicode[STIX]{x1D6E5}(G)$ is $k$-regular for some natural number $k$ if and only if $\overline{\unicode[STIX]{x1D6E5}}(G)$ is a regular bipartite graph.

scholarly journals Solvable groups whose character degree graphs generalize squares

2020 ◽  
Vol 23 (2) ◽  
pp. 217-234
Author(s):  
Mark L. Lewis ◽  
Qingyun Meng
Keyword(s):  
Solvable Group ◽  
Solvable Groups ◽  
Character Degree ◽  
Sylow Subgroups ◽  
Character Degree Graph ◽  
Product Of Subgroups ◽  
Degree Graph ◽  
Delta G ◽  
Definition Of ◽  
Square Graph

AbstractLet G be a solvable group, and let {\Delta(G)} be the character degree graph of G. In this paper, we generalize the definition of a square graph to graphs that are block squares. We show that if G is a solvable group so that {\Delta(G)} is a block square, then G has at most two normal nonabelian Sylow subgroups. Furthermore, we show that when G is a solvable group that has two normal nonabelian Sylow subgroups and {\Delta(G)} is block square, then G is a direct product of subgroups having disconnected character degree graphs.

Groups of automorphisms and centralizers

1990 ◽  
Vol 107 (2) ◽  
pp. 227-238 ◽  
Author(s):  
Alexandre Turull
Keyword(s):  
Solvable Group ◽  
Finite Solvable Group ◽  
Group Of Automorphisms ◽  
Fitting Height ◽  
Groups Of Automorphisms ◽  
The Difference

Let G be a finite solvable group and A a group of automorphisms of G such that (|A|, |G|) = 1. We denote by h(G) the Fitting height of G and by l(A) the length of the longest chain of subgroups of A. Then, under some additional hypotheses, it is known from [5] that h(G) ≤ 2l(A) + h(CG(A)) and from [8] that, when CG(A) = 1, h(G) ≤ l(A), both results being best possible (see [6, 7]). The present paper attempts to explain the difference in the coefficient of l(A) in the two inequalities, from 2 to 1.

scholarly journals Frobenius action on Carter subgroups

2020 ◽  
Vol 30 (05) ◽  
pp. 1073-1080
Author(s):  
Güli̇n Ercan ◽  
İsmai̇l Ş. Güloğlu
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Semidirect Product ◽  
Frobenius Group ◽  
Product Formula ◽  
Finite Solvable Group ◽  
Carter Subgroup ◽  
Text Length ◽  
Fitting Height ◽  
A Finite Group

Let [Formula: see text] be a finite solvable group and [Formula: see text] be a subgroup of [Formula: see text]. Suppose that there exists an [Formula: see text]-invariant Carter subgroup [Formula: see text] of [Formula: see text] such that the semidirect product [Formula: see text] is a Frobenius group with kernel [Formula: see text] and complement [Formula: see text]. We prove that the terms of the Fitting series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the Fitting series of [Formula: see text], and the Fitting height of [Formula: see text] may exceed the Fitting height of [Formula: see text] by at most one. As a corollary it is shown that for any set of primes [Formula: see text], the terms of the [Formula: see text]-series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the [Formula: see text]-series of [Formula: see text], and the [Formula: see text]-length of [Formula: see text] may exceed the [Formula: see text]-length of [Formula: see text] by at most one. These theorems generalize the main results in [E. I. Khukhro, Fitting height of a finite group with a Frobenius group of automorphisms, J. Algebra 366 (2012) 1–11] obtained by Khukhro.

MONOMIAL AND MONOLITHIC CHARACTERS OF FINITE SOLVABLE GROUPS

2021 ◽  
pp. 1-9
Author(s):  
BURCU ÇINARCI
Keyword(s):  
Solvable Group ◽  
Prime Divisor ◽  
Solvable Groups ◽  
Character Degree ◽  
Finite Solvable Group

Abstract Let G be a finite solvable group and let p be a prime divisor of $|G|$ . We prove that if every monomial monolithic character degree of G is divisible by p, then G has a normal p-complement and, if p is relatively prime to every monomial monolithic character degree of G, then G has a normal Sylow p-subgroup. We also classify all finite solvable groups having a unique imprimitive monolithic character.

BOUNDING THE NUMBER OF IRREDUCIBLE CHARACTER DEGREES OF A FINITE GROUP IN TERMS OF THE LARGEST DEGREE

2013 ◽  
Vol 13 (02) ◽  
pp. 1350096 ◽  
Author(s):  
MARK L. LEWIS ◽  
ALEXANDER MORETÓ
Keyword(s):  
Finite Group ◽  
Irreducible Character ◽  
Character Degree ◽  
Character Degrees ◽  
Character Degree Graph ◽  
Prime Factors ◽  
Degree Graph ◽  
A Finite Group

We conjecture that the number of irreducible character degrees of a finite group is bounded in terms of the number of prime factors (counting multiplicities) of the largest character degree. We prove that this conjecture holds when the largest character degree is prime and when the character degree graph is disconnected.

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